1. 1 Non-collinearity in high energy scattering processes Piet Mulders WHEPP X January 2008 mulders@few.vu.nl TITLE

2. 2 Outline Introduction: partons in high energy scattering processes (Non-)collinearity: collinear and non-collinear parton correlators –OPE, twist –Gauge invariance –Distribution functions (collinear, TMD) Observables –Azimuthal asymmetries –Time reversal odd phenomena/single spin asymmetries Gauge links –Resumming multi-gluon interactions: Initial/final states –Color flow dependence Applications Universality: an example gq  gq Conclusions OUTLINE

3. 3 QCD & Standard Model QCD framework (including electroweak theory) provides the machinery to calculate cross sections, e.g.  *q  q, qq   *,  *  qq, qq  qq, qg  qg, etc. E.g. qg  qg Calculations work for plane waves __ INTRODUCTION

4. 4 Confinement in QCD Confinement limits us to hadrons as ‘quark sources’ or ‘targets’ (with P X = P  p) These involve nucleon states At high energies interference terms between different hadrons disappear as 1/P 1.P 2 Thus, the theoretical description/calculation involves for hard processes, a forward matrix element of the form quark momentum INTRODUCTION

5. 5 Correlators in high-energy processes Look at parton momentum p Parton belonging to a particular hadron P: p.P ~ M 2 For all other momenta K: p.K ~ P.K ~ s ~ Q 2 Introduce a generic vector n ~ satisfying P.n = 1, then we have n ~ 1/Q, e.g. n = K/(P.K) Up to corrections of order M 2 /Q 2 one can perform the  -integration ~ Q~ M~ M 2 /Q INTRODUCTION

6. 6 (calculation of) cross section in DIS Full calculation + … + + + LEADING (in M/Q) INTRODUCTION

7. 7 (calculation of) cross section in SIDIS Full calculation + + … + + LEADING (in M/Q) INTRODUCTION

8. 8 Leading partonic structure of hadrons PHPH HPhPh pk h distribution correlator fragmentation correlator hard process Need P H.P h ~ s (large) to get separation of soft and hard parts Allows  d  … =  d(p.P)… PHPH PHPH  (x, p T ) PhPh PhPh  (z, k T ) INTRODUCTION

9. 9 Partonic correlators The cross section can be expressed in hard squared QCD-amplitudes and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (    or G). These are the (hopefully universal) objects we are after, useful in parametrizations and modelling. Fragmentation functions Distribution functions INTRODUCTION

10. 10 (non-)collinearity of parton correlators The cross section can be expressed in hard squared QCD-amplitudes and distribution and fragmentation functions entering in forward matrix elements of nonlocal combinations of quark and gluon field operators (    or G). These are the (hopefully universal) objects we are after, useful in parametrizations and modelling. lightcone lightfront:   = 0 TMD local Distribution functions collinear (NON-)COLLINEARITY

11. 11 Spin and twist expansion Local matrix elements in  Operators can be classified via their canonical dimensions and spin (OPE) Nonlocal matrix elements in  (x) Parametrized in terms of (collinear) distribution functions f…(x) that involve operators of different spin but with one specific twist t that determines the power of (M/Q) t-2 in observables (cross sections and asymmetries). Moments give local operators. Nonlocal matrix elements in  (x,p T ) Parametrized in terms of TMD distribution functions f…(x,p T 2 ) that involve operators of different spin and different twist. The lowest twist determines the operational twist t of the TMD functions and determines the power of (M/Q) t-2 in observables. Transverse moments give collinear functions. Spin n: ~ (P  1 …P  n – traces) Twist t: dimension  spin (NON-)COLLINEARITY

12. 12 Presence of gauge link needed for color gauge invariance The gauge link arises from all ‘leading’ m.e.’s as      Transverse pieces arise from A T   G  =  + A  + … Basic gauge links: Gauge invariance for quark correlators  [U - ] =  [  ]  [U + ] =  [  ] TIME RERVERSAL (NON-)COLLINEARITY

13. Distribution From  A T (  )  m.e. including the gauge link (in SIDIS) A+A+ One needs also A T G +  =  + A T  A T  (  )= A T  ( ∞ ) +  d  G +  Belitsky, Ji, Yuan 2002 Boer, M, Pijlman 2003 (NON-)COLLINEARITY

14. 14 Gauge invariance for gluon correlators Using 3x3 matrix representation for U, one finds in TMD gluon correlator appearance of two links, possibly with different paths. Note that standard field displacement involves C = C’ Basic gauge links (NON-)COLLINEARITY  g [  ]  g [  ]  g [  ]  g [  ]

15. 15 Collinear parametrizations Gauge invariant correlators  distribution functions Collinear quark correlators (leading part, no n-dependence) i.e. massless fermions with momentum distribution f 1 q (x) = q(x), chiral distribution g 1 q (x) =  q(x) and transverse spin polarization h 1 q (x) =  q(x) in a spin ½ hadron Collinear gluon correlators (leading part) i.e. massless gauge bosons with momentum distribution f 1 g (x) = g(x) and polarized distribution g 1 g (x) =  g(x) (NON-)COLLINEARITY

16. 16 TMD parametrizations Gauge invariant correlators  distribution functions TMD quark correlators (leading part, unpolarized) as massless fermions with momentum distribution f 1 q (x,p T ) and transverse spin polarization h 1  q (x,p T ) in an unpolarized hadron The function h 1  q (x,p T ) is T-odd! TMD gluon correlators (leading part, unpolarized) as massless gauge bosons with momentum distribution f 1 g (x,p T ) and linear polarization h 1  g (x,p T ) in an unpolarized hadron TMD correlators  q [U] and  g [U,U’] do depend on gauge links! (NON-)COLLINEARITY

17. unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity need p T T-odd (NON-)COLLINEARITY The quark distributions (in pictures)

18. 18 Results for deep inelastic processes OBSERVABLES DIS SIDIS

19. 19 Probing intrinsic transverse momenta In a hard process one probes quarks and gluons Momenta fixed by kinematics (external momenta) DIS SIDIS Also possible for transverse momenta SIDIS 2-particle inclusive hadron-hadron scattering Sensitivity for transverse momenta requires  3 momenta SIDIS:  * + H  h + X DY: H 1 + H 2   * + X e+e-:  *  h 1 + h 2 + X hadronproduction: H 1 + H 2  h 1 + h 2 + X  h + X (?) K2K2 K1K1      pp-scattering …and knowledge of hard process(es)! OBSERVABLES

20. 20 Time reversal as discriminator If time reversal can be used to restrict observable one has only even spin asymmetries If time reversal symmetry cannot be used as a constraint (SIDIS, DY, pp, …) one can nevertheless connect T-even and T-odd phenomena (since T holds at level of QCD). In hard part T is valid up to order  s 2 symmetry structure hermiticity parity time reversal combined OBSERVABLES

21. 21 Results for deep inelastic processes OBSERVABLES Sivers asymmetry Collins asymmetry Function as appearing in parametrization of  [+]

22. 22 Generic hard processes C. Bomhof, P.J. Mulders and F. Pijlman, PLB 596 (2004) 277 [hep-ph/0406099]; EPJ C 47 (2006) 147 [hep-ph/0601171] Link structure for fields in correlator 1 Matrix elements involving parton 1 and additional gluon(s) A + = A.n appear at same (leading) order in ‘twist’ expansion and produce link  [U] (1) insertions of gluons collinear with parton 1 are possible at many places this leads for correlator  (1) to gauge links running to lightcone ± infinity SIDIS   [+] (1) DY   [  ] (1) GAUGE LINKS

23. 23 Integrating  [±] (x,p T )   [±] (x) collinear correlator  [][] [][] GAUGE LINKS

24. 24 Integrating  [±] (x,p T )     [±] (x) transverse moments  G (p,p  p 1 ) T-evenT-odd Gluonic pole m.e. GAUGE LINKS

25. 25 A 2  2 hard processes: qq  qq E.g. qq-scattering as hard subprocess The correlator  (x,p T ) enters for each contributing term in squared amplitude with specific link  [Tr(U □ )U + ] =  [( □ )+] U □ = U + U  †  [U □ U + ] =  [ □ +] loop Traced loop GAUGE LINKS

26. 26 Gluonic poles Thus:  [U] (x) =  (x)   [U]  (x) =    (x) + C G [U]  G  (x,x) Universal gluonic pole m.e. (T-odd for distributions)  G (x) contains the weighted T-odd functions h 1  (1) (x) [Boer- Mulders] and (for transversely polarized hadrons) the function f 1T  (1) (x) [Sivers]   (x) contains the T-even functions h 1L  (1) (x) and g 1T  (1) (x) For SIDIS/DY links: C G [±] = ±1 In other hard processes one encounters different factors: C G [ □ +] = 3, C G [( □ )+] = N c Efremov and Teryaev 1982; Qiu and Sterman 1991 Boer, Mulders, Pijlman, NPB 667 (2003) 201 C. Bomhof, P.J. Mulders and F. Pijlman, EPJ C 47 (2006) 147 ~ ~ GAUGE LINKS

27. 27 examples: qq  qq in pp C G [D 1 ] D1D1 = C G [D 2 ] = C G [D 4 ] C G [D 3 ] D2D2 D3D3 D4D4 Bacchetta, Bomhof, Pijlman, M, PRD 72 (2005) 034030; hep-ph/0505268 APPLICATIONS

28. 28 examples: qq  qq in pp D1D1 For N c  : C G [D 1 ]   1 (color flow as DY) Bacchetta, Bomhof, D’Alesio,Bomhof, M, Murgia, PRL2007, hep-ph/0703153 APPLICATIONS

29. 29 Gluonic pole cross sections In order to absorb the factors C G [U], one can define specific hard cross sections for gluonic poles (which will appear with the functions in transverse moments) for pp: etc. for SIDIS: for DY: Bomhof, Mulders, JHEP 0702 (2007) 029 [hep-ph/0609206] (gluonic pole cross section) y APPLICATIONS

30. 30 Bomhof, M, Vogelsang, Yuan, PRD 75 (2007) 074019 Boer, M, Pisano, hep-ph/0712.0777 examples: qg  q  in pp D1D1 D2D2 D3D3 D4D4 Only one factor, but more DY-like than SIDIS Note: also etc. Transverse momentum dependent weigted APPLICATIONS

31. 31 Universality (examples qg  qg) D1D1 D2D2 D3D3 D4D4 D5D5 Transverse momentum dependent weighted UNIVERSALITY

32. 32 Universality (examples qg  qg) Transverse momentum dependent weighted UNIVERSALITY

33. 33 Universality (examples qg  qg) Transverse momentum dependent weighted UNIVERSALITY

34. 34 Universality (examples qg  qg) It is also possible to group the TMD functions in a smart way into two! (nontrivial for nine diagrams/four color-flow possibilities) But still no factorization! Transverse momentum dependent weighted UNIVERSALITY

35. 35 ‘Residual’ TMDs We find that we can work with basic TMD functions  [±] (x,p T ) + ‘junk’ The ‘junk’ constitutes process-dependent residual TMDs The residuals satisfies   (x) = 0 and  G (x,x) = 0, i.e. cancelling k T contributions; moreover they most likely disappear for large k T definite T-behavior no definite T-behavior Bomhof, Mulders, Vogelsang, Yuan, NPB, hep-ph/0709.1390 UNIVERSALITY

36. 36 Conclusions Beyond collinearity many interesting phenomena appear For integrated and weighted functions factorization is possible (collinear quark, gluon and gluonic pole m.e.) Accounted for by using gluonic pole cross sections (new gauge-invariant combinations of squared hard amplitudes) For TMD distribution functions the breaking of universality can be made explicit and be attributed to specific matrix elements Many applications in hard processes. Including fragmentation (e.g. polarized Lambda’s within jets) even at LHC References: Qiu, Vogelsang, Yuan, hep-ph/0704.1153 Collins, Qiu, hep-ph/0705.2141 Qiu, Vogelsang, Yuan, hep-ph/0706.1196 Meissner, Metz, Goeke, hep-ph/0703176 Collins, Rogers, Stasto, hep-ph/0708.2833 Bomhof, Mulders, hep-ph/0709.1390 Boer, Bomhof, Hwang, Mulders, hep-ph/0709.1087 CONCLUSIONS